DOCSLIB.ORG

The Jordan Curve Theorem 2

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Jordan Curve Theorem 2 GENERAL I ARTICLE The Jordan Curve Theorem 2. Conclusions Ritabrata Munshi Introd uction In the first part of the article (Resonance, Vol. 4, No.9 ) we proved the Jordan sepa.ration theorem which says that a simple closed curve in E2 separates it into at least two components. In this concluding part after some preliminary results we will prove the nonseparation theorem which states that an arc does not separate E2 and then put together Ritabrata Munshi is doing his M.Stat in the Indian all the pieces to conclude the proof of the Jordan Curve Statistical Institute, theorem. Calcutta. Apart from mathematics, he likes Prelhninary Lemmas painting and reading. We had defined when an arc is said to cross a circle. We Unlike most others he broaden the definition of crossing as follows: dislikes computers. Definition: Suppose f is a piece-wise circular simple closed curve and, is a piece-wise circular arc. Suppose we have the following situation: There exist PI, P2, P3, P4 E 'Y such that 'Y[P,P2] lies outside rand '[P:lP4] lies inside f while 'Y[P2,P3] is a subset of r (recall that JeT is true for r). Then we consider '[P2,P:l] as a crossing (see Figure 1). It could happen that P2 = P3 in which case we are in the context of the earlier definition of crossing. Lemma 2.1. Let 'Y, r be as above and p, q E 'Y such that P lies inside rand q lies outside r; then there is at least one crossing between l' and q. Proof. Suppose 'Y : [0,1] -t E2, the map corresponding to 'Y. Let 'Y-1(1') rand ,-l(q) = s; without loss, assume Figure 1. = r < s. Let 0' = sup{x ::; s\'Y(x) E inside (f)}; then obviously 'Y ( a.) E rand ,(0: - f) lies inside of r for -14--------------------------------~--------------------------------- RESONANCE , November 1999 GENERAL I ARTICLE small f > O. Also, as q lies outside (f) there exists 5 such that I'(x) E r for 0. ~ X ~ 5 and 'Y(5 + E) E outside (f) for small f > O. Hence 'Y [0.,5] is a crossing. So to 'move' from one component of (E2_f) to the other we have to cross f somewhere in between, and conversely, when­ ever we cross f we arrive at the other component. Hence Lemma 2.2. Let r be a piece-wise circular, simple, closed, curve, 'Y a piece-wise circular arc and a, b, the endpoints of 1'. If 'Y crosses r an odd number of times then a and bare in different components of (E2 - f). Suppose I' is a polygonal arc and C is a circle, such that 'Y crosses C at least once and the end points, a and b, of I' lie 'outside' C. Let 'Ye and 'Yo be the exterior paths of 'Y w.r.t. C. (In the Part 1 of this article we had defined 'Ye and 1'0 when 'Y is a polygonal line. However, the same prescription works in the case when 'Y is a piece-wise circular arc.) Let 5e := ('Ye - 'Y) and 00 := (1'0 - 1'), so 5e and 0'0 are subsets of C. In fact 5e and 5'0 are finite disjoint union of open arcs of C. Let II, ,In be the sequence in which the (open) arcs are visited while moving from a to b; then 5e = Uk=lIk. Similarly, 50 = UbI Jl· Lemma 2.3. Let p, q E oe and 0. be one of the arcs of C joining p and q. Then 0. contains an even number of crossings. Proof. First note that as a, b lie outside C, the total number of crossings in C is even. If p, q E Ik for some k, then number of crossings is zero in one of the arcs of C and hence the result is true in this case. N ow consider pElk, q E I k+ 1. If I k and I k+ 1 are separated by a point of touching (which is not a crossing) then the number of crossings is again zero in one direction. In the other case, let 'Y(P/,ql] be the part of 'Y traveled while moving from Ik to Ik+1. Assume 'Y(p',q') n C = ¢ for the time being. If the direction at p is same as that at q then form the piece-wise circular simple closed curve 'Y[p',q'] Up'pqq' and call if ~. Now observe that exactly two situations can arise -R-ES-O-N-A-N-C-E--I-N-o-v-e-m-b-er--19-9-9-----------~--------------------------------'5 GENERAL I ARTICLE depending on whether C - p'pqq' lies outside or inside of (~): (i) a, b are both inside (~) (Figure 2) or (ii) a, b are both outside (~) (Figure 3). Observe that in both cases the number of crossings in p'pqq' is even. A similar argument proves the lemma if the directions at p Figure 2. and q are different. Now if I(p'q) n C ¥= cp, then l'(p',q'] u p'pqq' will be a union of finite number of piece-wise circular simple closed curves. Then we can concentrate on each and every closed curve separately and prove the above claim. So the statement is true for p E Ik and q E Ik+l. It is easy to see that this implies the general statement! (Why?) Corollary 1. Suppose p E Dc, q E Dc and Q is an arc of C joining p and q. Then Q ,contains an odd number of crossings. Proof. It is enough to consider the case p E hand q E J1. And in this case a (the point where f hits the circle first) is the only crossing. Corollary 2. oc U Oc = cp. A Duality Result: Suppose TJ and ~ are two piece-wise circular simple closed curves and PI, P2 E TJ n ~ such that TJ 'crosses' ~ at TJ[Pl,P2] (= ~[Pl'P2])' then ~ also 'crosses' TJ at ~[Pl,P2]. (The notation TJ[pI,P2] and ~[Pl'P2] are ambiguous, because in each case there are two distinct arcs joining PI and P2. But here we take the arc common to TJ and ~ i.e. Figura 3. TJ[PIP2] = ~ (PIP2] . Also we have not defined crossing in this case; but the definition is obvious.) Proof. To prove duality we take sufficiently 'thin' strip neighbourhoods of TJ and ~ respectively and argue. Now we state and prove the main lemma of this section. Lemma 2.4. Let TJ be an arc joining P E OC, q E Oc and -16~~~------------~--------~-~------~------------------~---- RESONANCE I November 1999 GENERAL I ARTICLE satisfying 1] n 'Y = cp, then diam (1]) ~ min{d(a, C), d(b, Cn. Proof. Let 0: be an arc of C joining p and q. Let Rt, R2, ,R2s+1 be the crossings (of'Y and C) on 0:. Let Ca, Cb be the circles passing through a, b, respectively, concentric with C. We define a relation '-)0' on the set {Ri};!t1 by Ri --+ Rj if 'Y[R.i,Rj] does not interest CIl and Ca. Without loss, assume radius (Cb) ::; radius (Oa). It is easy to prove that '-)0' is an equivalence relation. It partitions the set {Rt, ,R2s+1} into equivalence classes. At least one of these equivalence classes has an odd number of points. Let {Ril' , Ri2J..+l} be one of them. Let E, F E Cb such that Ril' , Ri2>'+1 E 'Y(E,F) and 'Y(E,F) n Cb = ¢. Form ~ = 'Y[E,F] U EF (EF is anyone of the arcs of Cb joining E and F.) Now ()i crosses it an odd number of times, hence p and q lie in different components. That 1] is an arc joining p and q implies ~ U 1] =1= ¢ but 1] n'Y = cp => 1] U E F =1= ¢ =} 1] U Cb =1= cp. Also {p, q} C 1] U C, hence diam(1]) ~ (radius Cb - radius C) = min{ d(a, e), d(b, e)}. The N onseparation Theorem Suppose 'Y : [0,1) '-)0 E2 is an arc and a, b are two points in (E2 - 'Y)' We will prove that there exists an arc joining a and b, not intersecting 'Y. As the proof is a bit technical, we will first present the basic plan: Basic Plan: First we cover up 'Y with finite number of discs, this gives us a finite collection of circles. Suppose 1] in any piece-wise circular arc joining a and b, if it intersects 'Y then we can associate a number /-l(1], 'Y) with it, N ow we choose a suitable circle C from the collection and deform 1} to 1]C such that either 1]C does not intersect 'Y or -R-ES-O-N-A-N-C-E--I-N-o-v-em--be-r--19-9-9----------~~-----------------------------1-' GENERAL I ARTICLE where 8 is a fixed positive number. Then we choose a subarc of 'fie joining a and b, and continue the process. As 8 is a fixed positive number ~ after finite number of deformations we will get a piece-wise circular arc not intersecting" but joining a and b. N ow we give the proof in detail, Proof. Let a := min{d(a, ,), d(b,,)} > O. As, is a home­ Figure 4. omorphism, for each p E , there exists p', p" 3 P E , (p' ,pi') and diam (,[p',pll]) < a/2. (Of course, at the end points we have to take p = p' and p =1= p".) As E2 is regular, for each p E , there exists an open disc B (p, Tp) not intersecting the closed set ,[o,p'] U ,[P1l,1] U {a, b}.
Load more

Recommended publications
  • Geometric Algorithms
    Geometric Algorithms primitive operations convex hull closest pair voronoi diagram References: Algorithms in C (2nd edition), Chapters 24-25 http://www.cs.princeton.edu/introalgsds/71primitives http://www.cs.princeton.edu/introalgsds/72hull 1 Geometric Algorithms Applications. • Data mining. • VLSI design. • Computer vision. • Mathematical models. • Astronomical simulation. • Geographic information systems. airflow around an aircraft wing • Computer graphics (movies, games, virtual reality). • Models of physical world (maps, architecture, medical imaging). Reference: http://www.ics.uci.edu/~eppstein/geom.html History. • Ancient mathematical foundations. • Most geometric algorithms less than 25 years old. 2 primitive operations convex hull closest pair voronoi diagram 3 Geometric Primitives Point: two numbers (x, y). any line not through origin Line: two numbers a and b [ax + by = 1] Line segment: two points. Polygon: sequence of points. Primitive operations. • Is a point inside a polygon? • Compare slopes of two lines. • Distance between two points. • Do two line segments intersect? Given three points p , p , p , is p -p -p a counterclockwise turn? • 1 2 3 1 2 3 Other geometric shapes. • Triangle, rectangle, circle, sphere, cone, … • 3D and higher dimensions sometimes more complicated. 4 Intuition Warning: intuition may be misleading. • Humans have spatial intuition in 2D and 3D. • Computers do not. • Neither has good intuition in higher dimensions! Is a given polygon simple? no crossings 1 6 5 8 7 2 7 8 6 4 2 1 1 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 2 18 4 18 4 19 4 19 4 20 3 20 3 20 1 10 3 7 2 8 8 3 4 6 5 15 1 11 3 14 2 16 we think of this algorithm sees this 5 Polygon Inside, Outside Jordan curve theorem.
    [Show full text]
  • INTRODUCTION to ALGEBRAIC TOPOLOGY 1 Category And
    INTRODUCTION TO ALGEBRAIC TOPOLOGY (UPDATED June 2, 2020) SI LI AND YU QIU CONTENTS 1 Category and Functor 2 Fundamental Groupoid 3 Covering and fibration 4 Classification of covering 5 Limit and colimit 6 Seifert-van Kampen Theorem 7 A Convenient category of spaces 8 Group object and Loop space 9 Fiber homotopy and homotopy fiber 10 Exact Puppe sequence 11 Cofibration 12 CW complex 13 Whitehead Theorem and CW Approximation 14 Eilenberg-MacLane Space 15 Singular Homology 16 Exact homology sequence 17 Barycentric Subdivision and Excision 18 Cellular homology 19 Cohomology and Universal Coefficient Theorem 20 Hurewicz Theorem 21 Spectral sequence 22 Eilenberg-Zilber Theorem and Kunneth¨ formula 23 Cup and Cap product 24 Poincare´ duality 25 Lefschetz Fixed Point Theorem 1 1 CATEGORY AND FUNCTOR 1 CATEGORY AND FUNCTOR Category In category theory, we will encounter many presentations in terms of diagrams. Roughly speaking, a diagram is a collection of ‘objects’ denoted by A, B, C, X, Y, ··· , and ‘arrows‘ between them denoted by f , g, ··· , as in the examples f f1 A / B X / Y g g1 f2 h g2 C Z / W We will always have an operation ◦ to compose arrows. The diagram is called commutative if all the composite paths between two objects ultimately compose to give the same arrow. For the above examples, they are commutative if h = g ◦ f f2 ◦ f1 = g2 ◦ g1. Definition 1.1. A category C consists of 1◦. A class of objects: Obj(C) (a category is called small if its objects form a set). We will write both A 2 Obj(C) and A 2 C for an object A in C.
    [Show full text]
  • Lecture 1-2: Connectivity, Planar Graphs and the Jordan Curve Theorem
    Topological Graph Theory∗ Lecture 1-2: Connectivity, Planar Graphs and the Jordan Curve Theorem Notes taken by Karin Arikushi Burnaby, 2006 Summary: These notes cover the first and second weeks of lectures. Topics included are basic notation, connectivity of graphs, planar graphs, and the Jordan Curve Theorem. 1 Basic Notation graph: G denotes a graph, with no loops or multiple edges. When loops and multiple edges are present, we call G a multigraph. vertex set of G: denoted V (G) or V . The order of G is n = |V (G)|, or |G|. edge set of G: denoted E(G) or E. q = |E(G)| or kGk. subgraphs: H is a spanning subraph of G if V (H) = V (G) and E(H) ⊆ E(G). H is an induced subgraph of G if V (H) ⊆ V (G), and E(H) is all edges of G with both ends in V (H). paths: Pn denotes a path with n vertices; also called an n-path. cycles: Cn denotes a cycle with n vertices; also called an n-cycle. union of graph: G ∪ H has vertex set V (G) ∪ V (H) and edge set E(G) ∪ E(H). intersection of graph: G ∩ H has vertex set V (G) ∩ V (H) and edge set E(G) ∩ E(H). 2 Connectivity We begin by covering some basic terminology. k-connected: A graph G is k-connected if |G| ≥ k + 1 and for all S ⊆ V (G) with |S| < k, the graph G − S is connected. S separates G if G − S is not connected. cutvertex: v ∈ V (G) is a cutvertex if G − {v} is not connected.
    [Show full text]
  • How Twisted Can a Jordan Curve Be?
    HOW TWISTED CAN A JORDAN CURVE BE? Manfred Sauter Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless num- ber of possible exceptions. — Felix Klein (1849–1925) We start by fixing notation. A closed curve is a continuous map γ : [0, 1] → 2 R with γ(0) = γ(1), and it is a closed Jordan curve if in addition γ|[0,1) is injective. Similarly one defines a (Jordan) arc, a polygonal (Jordan) arc and a closed polygonal (Jordan) curve. It is well-known, of course, that a closed Jordan curve γ partitions the plane into three connected components: the bounded open interior int γ, the unbounded open exterior ext γ and the compact (image of the) curve itself, which is the common boundary of the first two components. For a captivating historical overview of the struggle to understand curves and toplological dimension we refer to [Cri99], where most of the classical topological results that appear in this note are featured. For a closed polygonal Jordan curve γ there is a simple criterion to check whether a point x ∈ R2 \ γ is in its interior. This criterion is algorithmic in nature and useful in computational geometry, see for example [O’R98, Section 7.4]. Consider a ray emerging from x towards infinity and count the number of intersections with γ. If the ray is chosen such that it intersects the segments of γ only ouside of their end points (which can be easily arranged for a polygonal Jordan curve) or if intersections with the end points are counted appropriately, the point x is in the interior of γ if and only if the number of intersections is odd.
    [Show full text]
  • Riemann's Mapping Theorem
    4. Del Riemann’s mapping theorem version 0:21 | Tuesday, October 25, 2016 6:47:46 PM very preliminary version| more under way. One dares say that the Riemann mapping theorem is one of more famous theorem in the whole science of mathematics. Together with its generalization to Riemann surfaces, the so called Uniformisation Theorem, it is with out doubt the corner-stone of function theory. The theorem classifies all simply connected Riemann-surfaces uo to biholomopisms; and list is astonishingly short. There are just three: The unit disk D, the complex plane C and the Riemann sphere C^! Riemann announced the mapping theorem in his inaugural dissertation1 which he defended in G¨ottingenin . His version a was weaker than full version of today, in that he seems only to treat domains bounded by piecewise differentiable Jordan curves. His proof was not waterproof either, lacking arguments for why the Dirichlet problem has solutions. The fault was later repaired by several people, so his method is sound (of course!). In the modern version there is no further restrictions on the domain than being simply connected. William Fogg Osgood was the first to give a complete proof of the theorem in that form (in ), but improvements of the proof continued to come during the first quarter of the th century. We present Carath´eodory's version of the proof by Lip´otFej´erand Frigyes Riesz, like Reinholdt Remmert does in his book [?], and we shall closely follow the presentation there. This chapter starts with the legendary lemma of Schwarz' and a study of the biho- lomorphic automorphisms of the unit disk.
    [Show full text]
  • Arxiv:1610.07421V4 [Math.AT] 3 Feb 2017 7 Need Also “Commutative Cubes” 12
    Modelling and computing homotopy types: I I Ronald Brown School of Computer Science, Bangor University Abstract The aim of this article is to explain a philosophy for applying the 1-dimensional and higher dimensional Seifert-van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the border between homology and homotopy. We explain some applications to filtered spaces, and special cases of them, while a sequel will show the relevance to n-cubes of pointed spaces. Key words: homotopy types, groupoids, higher groupoids, Seifert-van Kampen theorems 2000 MSC: 55P15,55U99,18D05 Contents Introduction 2 1 Statement of the philosophy3 2 Broad and Narrow Algebraic Data4 3 Six Anomalies in Algebraic Topology6 4 The origins of algebraic topology7 5 Enter groupoids 7 6 Higher dimensions? 10 arXiv:1610.07421v4 [math.AT] 3 Feb 2017 7 Need also “Commutative cubes” 12 8 Cubical sets in algebraic topology 15 ITo aqppear in a special issue of Idagationes Math. in honor of L.E.J. Brouwer to appear in 2017. This article, has developed from presentations given at Liverpool, Durham, Kansas SU, Chicago, IHP, Galway, Aveiro, whom I thank for their hospitality. Preprint submitted to Elsevier February 6, 2017 9 Strict homotopy double groupoids? 15 10 Crossed Modules 16 11 Groupoids in higher homotopy theory? 17 12 Filtered spaces 19 13 Crossed complexes and their classifying space 20 14 Why filtered spaces? 22 Bibliography 24 Introduction The usual algebraic topology approach to homotopy types is to aim first to calculate specific invariants and then look for some way of amalgamating these to determine a homotopy type.
    [Show full text]
  • 1 the Jordan Polygon Theorem
    Computational Topology (Jeff Erickson) The Jordan Polygon Theorem The fence around a cemetery is foolish, for those inside can’t get out and those outside don’t want to get in. — Arthur Brisbane, The Book of Today (1923) Outside of a dog, a book is man’s best friend. Inside of a dog, it’s too dark to read. — attributed to Groucho Marx 1 The Jordan Polygon Theorem The Jordan Curve Theorem and its generalizations are the formal foundations of many results, if not every result, in surface topology. The theorem states that any simple closed curve partitions the plane into two connected subsets, exactly one of which is bounded. Although this statement is intuitively clear, perhaps even obvious, the generality of the phrase ‘simple closed curve’ makes the theorem incredibly challenging to prove formally. According to most classical sources, even Jordan’s original proof of the Jordan Curve Theorem [3] was flawed; most sources attribute the first correct proof to Veblen almost 20 years after Jordan [6]. (But see also the recent defense and updated presentation of Jordan’s proof by Hales [2].) However, we can at least sketch the proof of an important special case: simple polygons. Polygons are by far the most common type of closed curve in practice, so this special case has immediate practical consequences. Moreover, most proofs of the Jordan Curve Theorem rely on this special case as a key lemma. (In fact, Jordan dismissed this special case as obvious.) 1.1 First, A Few Definitions 1 A homeomorphism is a continuous function h: X Y with a continuous inverse h− : X Y .
    [Show full text]
  • A Nonstandard Proof of the Jordan Curve Theorem
    Vladimir Kanovei,∗ Department of Mathematics, Moscow Transport Engineering Institute, Moscow 101475, Russia (email: [email protected] and [email protected]) Michael Reeken, Department of Mathematics, University of Wuppertal, Wuppertal 42097, Germany (email: [email protected]) A NONSTANDARD PROOF OF THE JORDAN CURVE THEOREM Abstract We give a nonstandard variant of Jordan's proof of the Jordan curve theorem which is free of the defects his contemporaries criticized and avoids the epsilontic burden of the classical proof. The proof is self- contained, except for the Jordan theorem for polygons taken for granted. Introduction The Jordan curve theorem [5] (often abbreviated as JCT in the literature) was one of the starting points in the modern development of topology (originally called Analysis Situs). This result is considered difficult to prove, at least compared to its intuitive evidence. C. Jordan [5] considered the assertion to be evident for simple polygons and reduced the case of a simple closed continuous curve to that of a polygon by approximating the curve by a sequence of suitable simple polygons. Although the idea appears natural to an analyst it is not so easy to carry through. Jordan's proof did not satisfy mathematicians of his time. On one hand it was felt that the case of polygons also needed a proof based on clearly stated geometrical principles, on the other hand his proof was considered in- complete (see the criticisms formulated in [12] and in [9]). If one is willing to assume slightly more than mere continuity of the curve then much simpler proofs (including the case of polygons) are available (see Ames [1] and Bliss [3] under restrictive hypotheses).
    [Show full text]
  • Invariance of Domain and the Jordan Curve Theorem in R2
    INVARIANCE OF DOMAIN AND THE JORDAN CURVE THEOREM IN R2 DAPING WENG Abstract. In this paper, we will first recall definitions and some simple ex- amples from elementary topology and then develop the notion of fundamental groups by proving some propositions. Along the way we will describe the alge- braic structure of fundamental groups of the unit circle and of the unit sphere, using the notion of covering space and lifting map. In the last three sections, we prove of the Jordan Separation Theorem, Invariance of Domain and finally 2 the Jordan Curve Theorem in R . Contents 1. Introduction 1 2. Homotopy and Fundamental Group 2 3. Covering Spaces and the Fundamental Group of S1 5 4. Induced Homomorphism and the No Retraction Theorem 7 5. Simple Connectedness of S2 9 6. The Jordan Separation Theorem 12 7. Invariance of Domain in R and R2 16 8. The Jordan Curve Theorem 19 Acknowledgments 22 References 22 1. Introduction Invariance of Domain and the Jordan Curve Theorem are two simply stated but yet very useful and important theorems in topology. They state the following: Invariance of Domain in Rn: If U is an open subset of Rn and f : U ! Rn is continuous and injective, then f(U) is open in Rn and f −1 : f(U) ! U is continuous, i.e. f is a homeomorphism between U and f(U). Jordan Curve Theorem in Rn: Any subspace C of Rn homeomorphic to Sn−1 separates Rn into two components, of which C is the common boundary. Though worded very simply, these two theorem were historically difficult to prove.
    [Show full text]
  • Section on the Van Kampen Theorem
    The Van Kampen theorem Omar Antol´ınCamarena Contents 1 The van Kampen theorem 1 1.1 Version for the full fundamental groupoid . 2 1.2 Version for a subset of the base points . 3 2 Applications 5 2.1 First examples . 5 2.1.1 The circle . 5 2.1.2 Spheres . 5 2.1.3 Glueing along simply-connected intersections . 5 2.1.3.1 Wedges of spaces . 6 2.1.3.2 Removing a point from a manifold. 6 2.1.3.3 Attaching cells. 6 2.1.3.4 Connected sums. 6 2.1.4 Compact surfaces . 6 2.2 Four dimensional manifolds can have arbitrary finitely generated fundamental group . 7 2.3 The Jordan Curve theorem . 7 2.3.1 The complement of a simple closed curve has exactly two components 8 2.3.2 A lemma about pushouts of groupoids . 9 2.3.3 The complement of an arc on S2 is connected . 10 1 The van Kampen theorem The van Kampen theorem allows us to compute the fundamental group of a space from infor- mation about the fundamental groups of the subsets in an open cover and their intersections. It is classically stated for just fundamental groups, but there is a much better version for fundamental groupoids: • The statement and proof of the groupoid version are almost the same as the statement and proof for the group version, except they're a little simpler! 1 • The groupoid version is more widely applicable than the group version and even when both apply, the groupoid version can be simpler to use.
    [Show full text]
  • 5 Brouwer's Fixed Point Theorem and the Jordan Curve Theorem
    5 Brouwer’s Fixed Point Theorem and the Jordan Curve Theorem 2 In this section we use the norm x = max x1 , x2 for x = (x1, x2) R and take as the unit | | {| | | |} ∈ circle the set S1 = x R2 / x = 1 with the usual topology inherited from R2. Thus S1 is { ∈ | | } the boundary of I2, where I = [0, 1] with the usual topology inherited from R, so it is actually in the shape of a square which is homeomorphic to our usual concept of a circle. For a function 2 f whose range is in R we denote by fi the ith coordinate function of f. Definition 5.1 A hex board is an array of regular hexagons arranged into a diamond shape in such a way that there is the same number of hexagons along each side of the board. The game of hex is played on a hex board in the following manner. Two players alternate and the aim of the game is to connect pairs of opposite sides. More precisely, if the players are labelled and then one pair of opposite sides is labelled and the other . In turn each player × ◦ × ◦ labels a previously unlabelled hexagon with her or his symbol. The winner is the player who first obtains a connected path of adjacent hexagons stretching between the sides of that player’s label. The picture below gives an example of a hex board in which there are six hexagons along each side. T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T Theorem 5.2 Every game of hex results in a winner.
    [Show full text]
  • The Isoperimetric Inequality in the Plane
    The isoperimetric inequality in the plane Andrew Putman Let γ be a simple closed curve in R2. Choose a parameterization f : [0, 1] → R2 of γ and define ∑k length(γ) = sup dR2 (f(ti−1), f(ti)) ∈ R ∪ {∞}; (0.1) ··· 0=t0<t1< <tk=1 i=1 here the supremum is taken over all partitions of [0, 1]. This does not depend on the choice of parameterization. By the Jordan curve theorem, the simple closed curve γ encloses a bounded region in R2; define area(γ) to be the Lebesgue measure of this bounded region. The classical isoperimetric inequality is as follows. Isoperimetric Inequality. If γ is a simple closed curve in R2, then area(γ) ≤ 1 2 4π length(γ) with equality if and only if γ is a round circle. This inequality was stated by Greeks but was first rigorously proved by Weier- strass in the 19th century. In this note, we will give a simple and elementary proof based on geometric ideas of Steiner. For a discussion of the history of the isoperimetric inequality and a sample of the enormous number of known proofs of it, see [1] and [3]. Our proof will require three lemmas. The first is a sort of “discrete” version of the isoperimetric inequality. A polygon in R2 is cyclic if it can be inscribed in a circle. Lemma 1. Let P be a noncyclic polygon in R2. Then there exists a cyclic polygon P ′ in R2 with the same cyclically ordered side lengths as P satisfying area(P ) < area(P ′). Proof.
    [Show full text]